Course: Trigonometry
Topic: The Six Trigonometric Functions
Subtopic: Basic Identities


Now that we have defined the six trig functions, we can begin to find relationships between them. These relationships are called identities. We will use these identities to simplify trigonometric expressions and to create even more identities. Simplifying a trig expression can be accomplished in many ways but the goal is to get it as simplified as possible.

The major techniques used to simplify a trig expression are:


By the end of this topic you should know and be prepared to be tested on:


Memorize the 3 reciprocal identities, the 2 ratio identities, and at least the first of the 3 Pythagorean identities. These are too important and used too often to just have on paper. Commit them to your memory!

Basic Trigonometric Identities to MEMORIZE

Reciprocal Identities

`csc theta = 1/sintheta`

`sec theta = 1/costheta`

`cot theta = 1/tantheta`

Ratio (a.k.a. Quotient) Identities

`tantheta = sintheta/costheta`

`cottheta = costheta/sintheta`

Pythagorean Identities




This diagram is a good way to visualize the three Pythagorean Identities. For instance by the Pythagorean Theorem on the middle triangle, do you see that `1 + (tantheta)^2 = (sectheta)^2`?

Angle in quadrant I forming three triangles having horizontal lengths cosine, 1, cotangent, heights sine, tangent, 1, and hypoteni 1, secant, cosecent, all respectively.

Mini-Lectures and Examples

STUDY: Basic Identities

Supplemental Resources (optional)

Videos from James Sousa's MathIsPower4U:
Mini-Lesson: Trig Identities:  Reciprocal, Quotient, Pythagorean
Mini-Lesson: Fundamental Identities:  Reciprocal, Quotient, Pythagorean

rev. 2021-04-05